Question

show all step please ((NOT in MATLAB))

except part d

Ralston's method

22.2 Solve the following problem over the interval from x = 0 to 1 using a step size of 0.25 where y(0) = 1. Display all your results on the same graph. Bolo dy bolo moto = (1 + 2x)VÝ Viena no woliszt unigas og tog (a) Analytically. sanoi solist og (b) Using Euler's method. (c) Using Heun's method without iteration. (d) Using Ralston's method. (e) Using the fourth-order RK method.

Answer 1

~ Given, dy (14 23 ) ) y(o)-1 = fode como (a) By Separation of variables, - ( 1+ 2x) dx Integrating on both sides. y = x+ 2C%")+c AL 2-0, ys 201) = 0+C - C-2 [rg = 1/₂ + 2% + 1] 10 Euler's method - Y 4 = 4; + h. f(x; 4;) y(0-25) = y(0)+ 0.25% f(0.4coo) where fox,y) = (1+2x) ry = 1+0.25x1) = 1.25 y(0.5) - y10.25) + 0.25% f0.25, 760.25)) = 1-25+ (0.25x1677) = 10669 YC0.75) = y(0.5) + 0.25x f(0.5, y(0.5)) - 1.669+ (0-25% 2.583) = 2.315 901) = ylo) y Co.75) + 0.25 f60.75, y Co.15)) 2:315+ (0.25% 3.804) = 3.266 . You = 3.266 using Euler's method.

(-) Heun's Method Git) = 4; + 1 / 2 (x + k ₂ ) where k= f(x: 14;) k=f(x,th, 9; +kh) From a Given, yo)=1. y (1)= yco) + - CK tk) - / k = fco.id=1 k = f(1, 1) = 3 = 1 + 1 (1+3) = 3 y)=3 by Heun's method.

(x) Using RK- 49th Order Method Vit1 = y; + (+2k, + 2k, + kuy? where k = f(x; 14;) & f ( zit h , y; + kb) = f (x + b it th ) k = f (x;th, 4; + k h) Giveni YCO)=t, fcx,y)= C1420)ry. y (0.25) = y(0)+ 0.25 ( k, + 2k + 2kg tk) k = f(0,1)= 1 k = f(0-125, 1-125) - 1.326 k.f (0.125, 1.166)= 1.3496 k=f(0125, 1.337) = 1.735 y(0.25)-1+0:25 (1+2C1-32. 1-3414 +*+35) = 1337 9(0-5) = 400-25) + 6.25 CK, A 2K,2x + k). k, f60-25,1-337) = 1.534 I 400.5) = 1.337 k = f(0.375,1.554)=2-181 + 0-25(1.734 +2 + K = f(0.375, 1.609) = 22202 +2(2-18H422 kg = f(0.5, 1.892)= 2.751 - 1.891

900:45) = 4(0-5)+ 0,25 (k;+ 23, + 25,31 kv) k= f(0.5, 1.891) 2.75 Iyco.45)=yco.5) + 0.887 k-f(0.625, 2.235) = 3.364 = 1.891+ 0.887 k f(0.625,23115) = 3:42) = 2778 k = f(1, 2.746) = 4.971 y (1)= y(0.75+ 0.25 (kit 2k + 2 kat ky) f(0.75,2478) = 4.167 Une u Co.15)+ 11264 0.042 x = f(0.875, 3.298) = 4.994 k = f(0.875, 3.402) = 5.072 ef (1,4 046) = 6.034. y)=4.042 by RK-49 Order Method